Projective general linear group action on the projective line
= Projective general linear group action on the projective line
The group $PGL_2(F)=GL_2(F)/Z$ acts faithfully on
$$
\mathbb P^1(F)=F\cup\{\infty\}
$$
by <Möbius transformations>. For $|F|=q$, this embeds $PGL_2(F)$ into the <symmetric group> $S_{q+1}$.